1. Field of the Invention
Attempts have also been made, with varying degrees of success, to apply the technique of Fourier transformation in the analysis of two-dimensional arrays of traveling waves, so that after the preparation of the original array it is possible to observe after suitable two-dimensional transformation into the frequency-wave number domain, the Fourier components of these various traveling waves. This, in turn, has led to the possibility of removing the effects of certain undesired waves by the elimination of the Fourier components of such waves from the transformed array, then re-transforming the remainder back to the original time-spacing domain.
Where reference is made to "two-dimensional data" or two-dimensional arrays, or the like, it should be understood how such phraseology is used throughout the specification and claims. In ordinary seismic prospecting, the received seismic waves at any geophone or geophone group represent (1) an amplitude, (2) the time after the source was actuated, and (3) a distance from source to receiver. Similarly, in seismic holography, the received seismic waves represent (1) an amplitude, (2) a phase relative to the source, and (3) two distances (ground coordinates) of receiver with respect to source. For the seismic reflection case, two-dimensional data refers to the fact that the usual visual presentation is amplitude in terms of the two dimensions of time and distance. In seismic holography, the visual presentation depends on both amplitude and phase in terms of the two dimensions of distance from the source. Accordingly, the two-dimensional data here are both amplitude and phase (usually presented as a single complex value) per data point, whose real and/or imaginary parts are plotted in terms of the two coordinate distances (such as N-S and E-W distances from source to receiver) defining each such data point. In both cases, the values at the data point are due to the passage of traveling seismic waves past the receivers.
It is to be understood that equivalent data are obtained similarly in other fields of physics, which can be similarly filtered by the method presented here.
2. Description of the Prior Art
One field in which there has been considerable application of spatial filtering techniques is in optical processing of seismic sections. Here the original array, which is to be Fourier-transformed into the frequency domain, is a two-dimensional matrix of real numbers. A photographic transparency of a seismic cross section is illuminated and the modulated light beam passed through a lens to form the Fourier transformation, which is filtered by masks in the Fourier plane, and the remainder transformed back.
For reference, see the article of P. L. Jackson, "Analysis of Variable-Density Seismograms by Means of Optical Diffraction", Geophysics, Vol. XXX, No. 1, p. 5 ff. He points out, for example, that illumination of a variable density transparency of the seismic section by a specially coherent monochromatic light source presents a visual object, a real two-dimensional array, and that, by passage of this image into a special optical system, it is possible to obtain at the principal focus of an objective lens the Fourier transform of the original real array, which is in a frequency-wave number domain, and therefore can be filtered. He shows various types of filters which are optical masks that can be employed in this optical arrangement, and the retransformed array after the optical filtration. This is only one of a number of articles on this subject. Additional references are found in the articles, "Velocity and Frequency Filtering of Seismic Data Using Laser Light", by Dobrin, Ingalls, and Long, Geophysics, XXX, No. 6, pp. 1144 ff, and "Optical Processing and Interpretation", by Fitton and Dobrin, Geophysics, Vol. XXXII, pp. 801 ff.
A related system for processing a seismic record section in such a way that seismic events with dips in a given range are preserved with no alteration over a wide frequency band, while seismic events with dips outside this specified range are severely attenuated, is described in the article, "Wide-Band Velocity Filtering--The Pie-Slice Process", by Embree, Burg, and Backus, Geophysics, Vol. XXVIII, No. 6, p. 948 ff. (See also U.S. Pat. No. 3,274,541 Embree.) In theory, this system relies on the conversion of seismic data obtained at the various geophones at differing distances from an impulsive source, by two-dimensional Fourier transformation, into the corresponding frequency and wave number (inverse of apparent wave length) plot. It was recognized that desired signals, i.e., those for reflected waves, differed in a systematic way from the low-velocity ground roll or high-velocity noise. In effect, the desired signals lay in one part of the f-k (frequency-wave number) plot of the data, and various kinds of noise lay in other sectors. Thus, by elimination of data from these other sectors and re-transforming, the resultant plot should involve mostly the signal range desired. This required for implementation a method in which, first, a plurality of reproducible input data consisting of seismic traces was obtained at different offset distances from the source; second, several of these input traces were individually filtered through filters, the individual filter being appropriately designed for the particular trace (corresponding to a particular offset distance from source to geophone); and, third, the responses of the plurality of filtered input traces was added together to form one equivalent output trace. This process was then repeated, leaving out one of the previously used input traces and adding another input trace, again using the appropriate filters, etc. This process is shown rather well on page 952 of the Embree, Burg, and Backus reference. Corresponding theory is given in the appendix to this reference. There it shows how the filters are designed in either in the time domain or in the frequency-wave number domain.
This sort of arrangement is at best only an approximation to the simultaneous use of all input data in a particular geophone array, and applying to the Fourier transform of these data an appropriate filter. This latter process can be accomplished using my invention, even for a relatively large amount of data.
Another point which should be made is that it is by now well understood that filtering, as the term is generally used, can be accomplished in quite a number of ways which are basically alternate schemes but which are all related. This is pointed out, among other places, by Mark Smith, in his article, "A Review of Methods of Filtering Seismic Data", which appeared in Geophysics, Vol. XXIII, No. 1 (January 1958), at pp. 44, et seq.
Other references could be given. However, the above is enough to illustrate the principles.
It is also known that it is possible to represent the response of a two-dimensional array of detectors to a steady state excitation from a source at an arbitrarily determined location. The data can be shown in a geometrically similar two-dimensional array of a set of numbers, in which the numbers assigned to any one point (corresponding geometrically to the location of the detector) represent the amplitude and phase response of the geophone to the impressed signal. Since this is steady state, simple harmonic motion, these numbers can be complex. The array shows the waves traveling directly from the known source to the particular detector, but also shows other waves which are of the same frequency as that of the source due to virtual sources (such as reflectors), diffraction from objects in the field, etc. This is seismic holography. See, for example, U.S. Pat. Nos. 3,400,363 and 3,461,420 Silverman, for seismic holographs. Photographs of real objects (not holograms) have been processed (filtered) after two-dimensional Fourier transformation. However, it should be pointed out that no reference has been found teaching how to apply such filtering technique to such data. The array, for example, can be analyzed by two-dimensional Fourier transformation, to separate out the effects of the virtual sources, diffraction, and the like.